I was immediately thinking about a two railway lane for the trolleys, with a pulley deep in the mountains and a long rope connection between the trolleys.

If you're digging upward, you can fill the trolley with stones. It will run down totally alone, pulling the rope behind, which goes over the pulley, and drags an empty car in. Or new workers or food. Anything which is lighter than the outgoing trolley.

Now ropes in the 14th century can't be extremely long. So you will have to build stations, where the full cart is attached to the next part of the railway. Like that a single cart full of stones on it's way out can pull in several carts, each one only to the next station.

Those stations need to be manned to handle the trolleys, to do the maintenance on the railway and pulleys and ropes. A city begins.

Furthermore, the dwarfs living in the mountains are forced to fill the trolleys with stones, always, or the railway stops working. This might be one good reason to continue building even when some groups of dwarfs say that the project will not be finished in their lifetime so you can as well let it be... no. You need to send stones downward to keep the railway working, always, or there is no supply.

That is, until they hit an underground water current. Then they can fill carts with water when sometimes they didn't make enough stones. This can be a great reason for fight between the dwarfs by the way, and a plotline driver. Also the other way round, when water accidentally fills an up going trolley and disrupts the system. Or when some part of the city needs water but gets stones sent... all kinds of nice plot drivers here.

In 13th century in Schwaz they also started to use water mills to keep the tunnels dry. The water would go over a big water wheel on it's way out of the tunnels, the wheel would provide force over transmission belts to some mechanics. Sometimes they used it to move trolleys upward, sometimes they hung buckets on it to get even more water or stones out of the tunnels below. In Schwaz they also used kids to get silver out of the mountain. They would be paid at the entrance with a loaf of bread when they came out with silver.

So. Sorry, no exquisite perfect answer here. But hopefully some ideas and some impressions. Make sure to notify me when your book is done, I'm a fan of fantasy dwarfs.

Start with a natural cave system

Many natural cave systems are hundreds of km long. Mammoth has over 600 km of explored cave systems and at its widest is at least 20-30km however no one really knows for sure since there are tons of unexplored tunnels meaning it could be much much bigger.

While this may not give you 64km as the crow flies, there are plenty of natural caves in mountainous regions than take a walk of more than 64km from point A to point B. Finding one that goes under a mountain that is too hard to go over is quite possible.

At first, it needn't be anything more than a wide tunnel that allows four carriages to pass through (let's say 12 meters wide and 4-5 meters tall)

This is a very unreasonable expectation in the 13th century whether this is man-made or not. Most roads built above ground in that time period were only 4-8 meters wide because wagons were not as common back then as cars are today; so, there were few places where you would expect to need multiple lanes going in one direction.

Most medieval carts and wagons were built to the same width as old Roman carts and wagons so that they could use the same roads. This width was about 1.5 meters. Since underground roads are much more expensive, you won't waste a lot of money on extra room; so, a more likely width for two way cart traffic is going to be something like 4m wide and 4m tall, but to really get started you don't need even that much. A donkey train can go over much rougher terrain and through passages as small as about 2.5m by 2.5m giving you enough room for 2 way traffic. So, your initial usage of the passage will probably be donkeys since you will barely need to modify the caves at all for that. As time goes on you can carve things bigger and smoother until it accommodates a 4x4 space all the way through, then scale that up to 12x4 only if the need arises (which it probably won't until the mid 1900s)

Because of the long trip, way stations may be installed to give travelers places to rest. Over time the quality of roads will improve and new joining tunnels will be dug to make the trip shorter, safer, and easier, and one or more way stations may grow into a proper town in its own right collecting money from travelers as they pass through to guide them safely through the massive underground labyrinth, getting paid by the local lords to expand the tunnels, and not to mention whatever natural resources the mountains already contain.

As waystations grow into towns, ventilation will of course have to be solved for, but the locations of these waystations will likely be chosen as places that have easy access to sunlight, water, and air anyway. So, if Moria is near the surface but sandwiched between two mountain ridges, it could get easy access to above ground resources despite being a long trip from either side.

Hankrecords brings up a good question in the comments about how Moria's location would be planned, and the answer lies in the origin of the cave system itself. Many caves are created from aquifers eroding as they drain out. Basically you start off with an elevated lake or lakes sandwiched between the mountain ridges, as water drains off through underground aquifers it will erode the ground away following the path of the softest minerals creating underwater caves. Eventually these caves will begin to break through to the surface at lower elevations allowing the lake and caves to drain out.

This will create a natural sinkhole entrance where the lake used to be high above what will become the main town which will provide enough sunlight, water, and air for a significant underground population. The town may also have farms above ground in the valley that is otherwise inaccessible to the outside world.

Given the huge investment in terms of time, material and lives, the chances of an underground city developing because someone wanted a 60km tunnel built hrough a mountain range in a world with knowledge and tech similar to 14th century Europe are rather slim.

It would be more likely to develop such a city if you started off with a mine or a group of cave dwellers with belligrent neighbours who keep digging deeper tunnels/caves to stay safe than a tunnel project - i.e. like Moria, have some or other mineral that was discovered in a cave on the side of the mountain or in a streambed of a stream with its source underground in the mountain that is rare and precious enough that when the easily accessible mineral on or near the surface is mined out it would make sense to tunnel deeper. As the mines get deeper, workers start sleeping in the mine on their off shift and only returning to the surface once a week or whenever their "rest day" may be or if they are slaves to begin with, their "masters" may leave them in the mine for life. Either way, as the mines got deeper, making first overnight camps and eventually settlements underground may make sense. They could even cut through into natural cave systems which they decide will make a decent living space.

Cutting through the mountain then becomes a "byproduct " rather than an original goal. Maybe some slaves started a religion where the tunnel was needed (some day all our people will reach the promised land when the tunnel reaches that far, where we will be free) or after several generations of following mineral-bearing veins a mining tunnel eventually broke through to the other side. Once this happened, the possibilities of establishing a shorter trade route became possible, so the tunnels were enlarged, reinforced and cleaned up and only then did the underground village really develop (as a way station for travelers through the tunnels).

Edit: Another option is if you make the underground living sort of non-optional e.g. if your region has horrendous winters with snow meters thick, you might end up with a system where people have summer and winter homes farming, hunting, gathering etc. out on the plains & forest & fishing rivers&ocean then retreat to winter caves when the weather turns. For a civilisation that evolved that way, tunneling deeper to hollow out new homes would not be that far fetched. Also, with those types of conditions, they may well decide a tunnel through the mountain is a good idea, as they would be unable to travel any other way for half the year and would have time on their hands. This would be a generations project, though. They would work on it all winter, but summers would be needed for gathering food.

(convert to comment if not good enough)

You might have a look at De Re Metallica, which has detailed drawings and discussion of 1500s mining technology. OK, couple centuries later than you posit, but maybe 1500s would be better assumption than 1300s. There were bellows, waterwheels, etc. So, you can see some ability to go after what you are talking about.

I still sort of doubt the economic justification for it (even now, really, but definitely with 1300s or 1500s tech). But maybe with some ruler just forcing it to happen, could be done.

https://en.wikipedia.org/wiki/De_re_metallica

http://farlang.com/books/agricola-hoover-de-re-metallica

https://commons.wikimedia.org/wiki/Category:Illustrations_from_De_re_metallica_(1556)_png

Base it on a qanat.

Qanats are deep tunnels bored into mountains to convey water down to dry lowlands. They are built using ancient Persian tech which impressed the Romans when they showed up. 14th century European medievals would not have been able to do this. These tunnels serve their intended purpose to this day.

The Qanats of Iran, Scientific American 1968

The qanat system consists of under­ ground channels that convey water from aquifers in highlands to the surface at lower levels by gravity. The qanat works of Iran were built on a scale that rivaled the great aqueducts of the Roman Em­ pire. vVhereas the Roman aqueducts now are only a historical curiosity, the Iranian system is still in use after 3,000 years and has continually been expand­ ed. There are some 22,000 qanat units in Iran, comprising more than 170,000 miles of underground channels. The sys­ tem supplies 75 percent of all the water used in that country, providing water not only for irrigation but also for house­ hold consumption

The linked article provides details about how the qanats were dug. The coolest thing to me was sighting underground to keep the tunnel straight - they started at the exit and were trying to hit a well far inland. By far I mean a serious long way; the longest known qanat is 71 km.
https://en.wikipedia.org/wiki/Traditional_water_sources_of_Persian_antiquity

In your world the qanat is ancient. Possibly the construction techniques have been lost. The flowing water is key because that is what draws air down thru the vent shafts. There are methods used today to cool houses using ancient qanats - if you have a chimney it can create a suction and draw cool air up from the qanat flowing beneath the house.

Making side tunnels off the great qanat is a reasonable thing. Over time these tunnels grow. The main thing for this city is not to mess with the water - it has to flow or the city will suffocate.

Cappadocia is a mountainous region full of underground cities. The larges one is Derinkuyu, which was built some 700 or 800 hundred years BC and could house probably 20,000 people. Those guys were 22 centuries behind your world's people in technology and they managed it.

You could draw some inspiration from these underground cities. Do notice that this city was initially built by digging through soft volcanic rock; I am not sure ranges such as the Andes would be as easy to dig, but in a fictional world it may be a possibility.

In ad 41 the Romans used some 30,000 men for 10 years to push a 3.5-mile (6-kilometre) tunnel to drain Lacus Fucinus. (Source)

Unfortunately, 14th century Europe wasn't, from an engineering perspective, more advanced than 1st century Rome. So I consider that estimate to be reasonably valid.

• 30,000 people @ 60 km = 100 years.

Could you simply scale this? (300,000 people would take 10 years)? You need to be careful with that. There's only so much space inside a tunnel. The Romans were pretty good (not godlike, but pretty good) at maximizing their efficiency, so my gut tells me the best you could do by adding more people is to pull 100 years down to 80.

Also note that both medieval Europe and Rome have one thing in common: no explosives. The work would be brutal and you can bet your bottom dollar that a lot of those 30,000 men died during construction.

Problems

That same source makes the following observation about ancient tunnels:

Ventilation methods were primitive, often limited to waving a canvas at the mouth of the shaft, and most tunnels claimed the lives of hundreds or even thousands of the slaves used as workers.

Even by the 14th century, atmosphere, gasses, and human physiology were only crudely understood and what we, today, understand as completely obvious (duh!) was to them, far too often, incomprehensible witchcraft.

• Without massive ventilation, possibly requiring active fans, a 60km tunnel would kill most people before they ever got through it. You could possibly help this by making the tunnel consistently sloped, but that means there's a boat-load of bad effluvia at one end. Barometric changes at both ends will complicate this — and downhill or not, if those pressures line up correctly, the air inside your tunnel won't move an inch (everybody's dead).

• Average hiking speed is 2 miles (3.2 km) an hour. Your tunnel's basically 19 hours non-stop, which means people will stop. Possibly extending the passage to 3 or more days. During that time you have fires (camping, heat, and torches), defecating people, defecating animals, bats, dust and gas, all building up over time. See the previous point.

• Sewage removal could be a challenge, unless you get lucky and find a couple of cave systems along the way and some water sources to wash it all down the caves.

• Speaking of water... pray you don't find much.

• The tech existed to shore up the tunnels, but historically it wasn't used much to do so, so you'll have problems with collapses.

• Caves (tunnels...) are notoriously slippery and cold.

• Heaven help you if someone half-way through the trip goes mad from claustrophobia or Nyctophobia.

• And remember if you do make it slightly sloped, whomever controls the up-slope entrance has a decisive military advantage if they have access to a river.

A few centuries

By the middle ages, miners were quite good. They employed technology that was advanced for the time to solve major problems like flooding, ventilation, and removing spoil. If you're willing to wave your hands about some practical concerns, you could conceivably have your tunnel and underground living areas built within a few centuries. When you're thinking about staffing levels, remember that the Channel Tunnel involved 13,000 workers using 1980s tech, and that's not counting all the people who were tangentially involved (making food, manufacturing tunneling equipment, etc.). So 100,000 people is not an excessive number.

Challenge 1: Tunnel life

The realities of living in a tunnel would be harsh. You mention that the miners can see in the dark, which is important. They'll need fresh water brought in, waste removed, etc. You say that the living areas will be near the entrances to the tunnel, but it might be more useful to put the living areas near the middle. After all, your workers are going to be tired after swinging a pickax all day and will be grumpy if you make them walk or ride a mule cart for 25 miles to get back to the entrance housing. At that point, they might as well live outside.

Fortunately, you won't need to house 100,000 people in the tunnel. A large portion of your workers can stay outside. This project will require an entire city's worth of support activities. For example, you'll need animal breeders for fresh animals, blacksmiths to build and repair tools, lumberjacks to fell trees for support timbers, tailors to make clothing, laborers to dump the rock coming out of the mine, etc.

Challenge 2: Air

You'll need a supply of fresh air deep within the tunnel. There is precedent to mines installing air supplies in the middle ages. Here's a great guide to mining technology in the era you're considering. I suggest reading the whole thing, but here's a relevant excerpt:

Delving deeper beneath the surface of the earth led to a second complication for the miners: less oxygen for those working in the elongated tunnels. Rather than limit the depth that the mines could extend, ventilating machines were developed as a solution. The simplest form of ventilation, sufficing only for the shallower mines, was merely the flapping of cloths to circulate air. Later, revolving fans and single- or double-acting bellows maintained air flow, while allowing miners to dig to new depths.

So you can hand wave a superpowered bellows to keep air moving.

Challenge 3: Transportation

As the tunnel gets longer, it'll become harder and harder to get stuff in and out. Even with two lanes moving in each direction, horse-drawn railcars aren't fast. Let's assume you're running a really coordinated operation where each car is filled rapidly and there's always another one waiting right after it to take the next bit of spoil. You'll still need a lot of people running the carts. And remember that the carts will have to go far enough out of the tunnel to dump the rock, so by the end of the project, your carts could conceivably be traveling a 100-mile round trip. The good news is that you'll have lots of empty carts going into the tunnel to carry supplies.

Challenge 4: Precision

It's going to be really really hard to hit a specific target from one end after all that digging. Especially given that miners were used to digging paths that weren't necessarily straight. Here's a great description of a mine from Reddit:

How did a medieval mine look? Some were open pits, but most were mine shafts, dug down and reinforced with wooden supports. The miners tried to follow the streaks of ore in the mountain, and the mines could meander quite a bit. The larger mines had not only pumps keeping water out, but also bellows to pump air down to the miners. To break the mountain, the miners would light fires down in the mining shafts to heat the mountain, and then rapidly cool it by throwing water on it. It cracked, and was then much easier to pickaxe and shovel into small handcarts used to transport the ore to the shafts...

The big question:

Data from quanats suggests 2-5 metres per day. This does not account for the hard rock, but it also assumes relatively small teams and getting the spoils out through narrow shafts. Bigger tunnels have a bigger cross section, but they also allow larger teams to work at the same time. So call it 35-90 years for the full distance.

The little comments:

• Coming from one side only is no problem. There is no real chance to have two opposite tunnels meet in the middle. They might miss each other by miles and both tunnel the entire way.
• Similarly, they could not really predict how the terrain on the other side looks because they can't really align the maps on both sides of the range.
• Growing food underground sounds pointless. Transporting it 40+ miles is easier than growing underground.
• Ventilation will be a big issue. Do they dig vertical airshafts every couple of hundred yards or do they use forced horizontal ventilation?
  • Bellows or pumps operated by men, animals, waterwheels?
  • Suction through either a fire or moving water?
• Light will have to be flame, which further complicates the air supply. If the pumps fail for a few hours, people might die.
• Transport of first spoils and food and then cargo with horse-drawn carts on wooden rails?